Abstract
Chapter 4 is an introduction to the finite element and finite volume methods for scalar transport and kinematic waves. The Galerkin method is developed based on the steady-state diffusion equation, and extended to time dependent problems by means of standard methods of solving systems of ordinary differential equations. The method's inability to yield non-oscillatory results for advection and kinematic waves is used as a motivation for the derivation of the Petrov-Galerkin and characteristic-Galerkin methods. The analysis is used to stress the difference between local and global conservation. By careful selection of trial functions, quadrature methods, and filtering schemes, the relation among finite-difference, element and volume methods is revealed. The QUICK scheme for advection is constructed, and total variation methods are introduced along with the concept of flux limiting.
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